FRANÇOIS-SERGE LHABITAN T 195with a view toward developing a theory that accounts for market imperfec-
tions (for example, transaction costs, liquidity problems, feedback effects,
etc.). The second direction focused on generalizing the price dynamics of
the traded assets to include broader classes of stochastic processes, such as
the so-called Levy processes and their extensions. Finally, the third direc-
tion started considering the case of more complex financial products, such
as exotic options and structured products. In each of these directions, new
quantitative models and techniques have been developed and applied, and
this trend towards more financial engineering is likely to persist. There is
no turning back, despite the fact that quantitative finance is regularly pro-
nounced dead, particularly after major market events such as the crashes of
1987, 1994 or 1998.
Unfortunately, the level of complexity in the new financial models is also
rising. The new science behind quantitative finance is relying more and more
on applied probability theory and numerical analysis, and uses mathemati-
caltechniquessuchasstochasticcalculusandpartialdifferentialequationsto
achieve its results. It also draws on wide areas of physics, notably heat diffu-
sion and fluid mechanics where the dynamics are similar to those of financial
markets.^1 As a side effect, not surprisingly, the mathematics encapsulating
many of the more innovative derivatives is less and less accessible to the
majorityofmarketparticipants, includingseniormanagement. Thisresulted
inthecreationofaseriesofnewmodelswhoseroleistohelpseniormanagers
understanding the risks of instruments that are themselves heavily depen-
dent on models. It is a vicious circle, which was summarized as follows by
AlanGreenspaninoneofhisallocutionsinMarch1995: “Thetechnologythat
is available has increased substantially the productivity for creating losses”,
and empirical evidence showed that these losses could be significant.^2
10.3 An illustration of model risk
The concept of implied volatility is probably among the best illustrations of
what we mean by model risk. Consider for instance the case of the Black
and Scholes option-pricing framework. The price of an option depends
upon the price of the underlying asset, the exercise price, the time to matu-
rity, the future volatility of the underlying asset, and the risk-free interest
rate. All these parameters are observable, except the volatility parameter,
which needs to be estimated. Now, how can one estimate something that is
unobservable?
The usual way to price an option is to plug a volatility estimate into a pric-
ing model (as well as all the others necessary observed variables) and thus
obtain the corresponding option price. Alternatively, one can also “invert
the model”. Starting from a market-quoted option price, one can compute
the implied volatility assuming that the Black-Scholes formula is the correct